A dual-dual mixed formulation for nonlinear exterior transmission problems

Gatica, Gabriel; Meddahi, Salim

doi:10.1090/S0025-5718-00-01267-9

A dual-dual mixed formulation for nonlinear exterior transmission problems
HTML articles powered by AMS MathViewer

by Gabriel N. Gatica and Salim Meddahi PDF

Math. Comp. 70 (2001), 1461-1480 Request permission

Abstract:

We combine a dual-mixed finite element method with a Dirichlet-to-Neumann mapping (derived by the boundary integral equation method) to study the solvability and Galerkin approximations of a class of exterior nonlinear transmission problems in the plane. As a model problem, we consider a nonlinear elliptic equation in divergence form coupled with the Laplace equation in an unbounded region of the plane. Our combined approach leads to what we call a dual-dual mixed variational formulation since the main operator involved has itself a dual-type structure. We establish existence and uniqueness of solution for the continuous and discrete formulations, and provide the corresponding error analysis by using Raviart-Thomas elements. The main tool of our analysis is given by a generalization of the usual Babuska-Brezzi theory to a class of nonlinear variational problems with constraints.

References

Gabriel R. Barrenechea, Gabriel N. Gatica, and Jean-Marie Thomas, Primal mixed formulations for the coupling of FEM and BEM. I. Linear problems, Numer. Funct. Anal. Optim. 19 (1998), no. 1-2, 7–32. MR 1606917, DOI 10.1080/01630569808816812
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
Ulrich Brink, Carsten Carstensen, and Erwin Stein, Symmetric coupling of boundary elements and Raviart-Thomas-type mixed finite elements in elastostatics, Numer. Math. 75 (1996), no. 2, 153–174. MR 1421985, DOI 10.1007/s002110050235
Carsten Carstensen and Ernst P. Stephan, Adaptive coupling of boundary elements and finite elements, RAIRO Modél. Math. Anal. Numér. 29 (1995), no. 7, 779–817 (English, with English and French summaries). MR 1364401, DOI 10.1051/m2an/1995290707791
Martin Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (1988), no. 3, 613–626. MR 937473, DOI 10.1137/0519043
Martin Costabel and Ernst P. Stephan, Coupling of finite and boundary element methods for an elastoplastic interface problem, SIAM J. Numer. Anal. 27 (1990), no. 5, 1212–1226. MR 1061127, DOI 10.1137/0727070
M. Feistauer, Mathematical and numerical study of nonlinear problems in fluid mechanics, Equadiff 6 (Brno, 1985) Lecture Notes in Math., vol. 1192, Springer, Berlin, 1986, pp. 3–16. MR 877102, DOI 10.1007/BFb0076047
Miloslav Feistauer, On the finite element approximation of a cascade flow problem, Numer. Math. 50 (1987), no. 6, 655–684. MR 884294, DOI 10.1007/BF01398378
Kang Feng, Finite element method and natural boundary reduction, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 1439–1453. MR 804790
G.N. Gatica, On the Coupling of Boundary Integral and Finite Element Methods for Nonlinear Boundary Value Problems, Ph.D. Dissertation, University of Delaware, (1989).
G. N. Gatica, Combination of mixed finite element and Dirichlet-to-Neumann methods in nonlinear plane elasticity, Appl. Math. Lett. 10 (1997), no. 6, 29–35. MR 1488266, DOI 10.1016/S0893-9659(97)00101-8
G.N. Gatica, Solvability and Galerkin approximations of a class of nonlinear operator equations, Technical Report 99-03, Departamento de Ingeniería Matemática, Universidad de Concepción, (1999). http://www.ing-mat.udec.cl/inf-loc-dim.html
G.N. Gatica and N. Heuer, Minimum residual iteration for a dual-dual mixed formulation of exterior transmission problems, Technical Report 99-07, Departamento de Ingeniería Matemática, Universidad de Concepción, (1999). http://www.ing-mat.udec.cl/inf-loc-dim.html
G.N. Gatica and N. Heuer, An expanded mixed finite element approach via a dual-dual formulation and the minimum residual method, Technical Report 99-08, Departamento de Ingeniería Matemática, Universidad de Concepción, (1999). http://www.ing-mat.udec.cl/inf-loc-dim.html. To appear in Journal of Computational and Applied Mathematics.
G.N. Gatica and N. Heuer, Conjugate gradient method for dual-dual mixed formulations, Technical Report 99-16, Departamento de Ingeniería Matemática, Universidad de Concepción, (1999). http://www.ing-mat.udec.cl/inf-loc-dim.html
G. N. Gatica and G. C. Hsiao, On a class of variational formulations for some nonlinear interface problems, Rend. Mat. Appl. (7) 10 (1990), no. 4, 681–715 (1991) (English, with Italian summary). MR 1145674
Gabriel N. Gatica and George C. Hsiao, On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in $\textbf {R}^2$, Numer. Math. 61 (1992), no. 2, 171–214. MR 1147576, DOI 10.1007/BF01385504
Gabriel N. Gatica and George C. Hsiao, Boundary-field equation methods for a class of nonlinear problems, Pitman Research Notes in Mathematics Series, vol. 331, Longman, Harlow, 1995. MR 1379331
Gabriel N. Gatica and George C. Hsiao, The uncoupling of boundary integral and finite element methods for nonlinear boundary value problems, J. Math. Anal. Appl. 189 (1995), no. 2, 442–461. MR 1312055, DOI 10.1006/jmaa.1995.1029
Gabriel N. Gatica and Wolfgang L. Wendland, Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems, Appl. Anal. 63 (1996), no. 1-2, 39–75. MR 1622612, DOI 10.1080/00036819608840495
Nelson Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635–646. MR 98
V. Girault and P.A. Raviart, Finite Element Approximation of the Navier-Stokes Equations: Theory and Algorithms, Springer, Berlin Heidelberg New York, 1986.
Dan Givoli, Numerical methods for problems in infinite domains, Studies in Applied Mechanics, vol. 33, Elsevier Scientific Publishing Co., Amsterdam, 1992. MR 1199563
Hou De Han and Xiao Nan Wu, The approximation of the exact boundary conditions at an artificial boundary for linear elastic equations and its applications, Math. Comp. 59 (1992), no. 199, 21–37. MR 1134732, DOI 10.1090/S0025-5718-1992-1134732-0
H. Han and W. Bao, The artificial boundary conditions for incompressible materials on an unbounded domain, Numer. Math. 77 (1997), no. 3, 347–363. MR 1469676, DOI 10.1007/s002110050290
Bodo Heise, Nonlinear field calculations with multigrid-Newton methods, Impact Comput. Sci. Engrg. 5 (1993), no. 2, 75–110. MR 1223880, DOI 10.1006/icse.1993.1004
Bodo Heise, Analysis of a fully discrete finite element method for a nonlinear magnetic field problem, SIAM J. Numer. Anal. 31 (1994), no. 3, 745–759. MR 1275111, DOI 10.1137/0731040
George C. Hsiao and Shangyou Zhang, Optimal order multigrid methods for solving exterior boundary value problems, SIAM J. Numer. Anal. 31 (1994), no. 3, 680–694. MR 1275107, DOI 10.1137/0731036
Rainer Kress, Linear integral equations, Applied Mathematical Sciences, vol. 82, Springer-Verlag, Berlin, 1989. MR 1007594, DOI 10.1007/978-3-642-97146-4
Salim Meddahi, An optimal iterative process for the Johnson-Nedelec method of coupling boundary and finite elements, SIAM J. Numer. Anal. 35 (1998), no. 4, 1393–1415. MR 1620160, DOI 10.1137/S0036142996300762
Salim Meddahi, Javier Valdés, Omar Menéndez, and Pablo Pérez, On the coupling of boundary integral and mixed finite element methods, J. Comput. Appl. Math. 69 (1996), no. 1, 113–124. MR 1391614, DOI 10.1016/0377-0427(95)00023-2
Jindřich Nečas, Introduction to the theory of nonlinear elliptic equations, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1986. Reprint of the 1983 edition. MR 874752
J.E. Roberts and J.-M. Thomas, Mixed and Hybrid Methods, in Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.L. Lions, vol. II, Finite Element Methods (Part 1), 1991, North-Holland, Amsterdam.
E. P. Stephan, Coupling of finite elements and boundary elements for some nonlinear interface problems, Comput. Methods Appl. Mech. Engrg. 101 (1992), no. 1-3, 61–72. Reliability in computational mechanics (Kraków, 1991). MR 1195578, DOI 10.1016/0045-7825(92)90015-C
A. Ženíšek, Nonlinear elliptic and evolution problems and their finite element approximations, Computational Mathematics and Applications, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1990. With a foreword by P.-A. Raviart. MR 1086876